where \(\mathbin {\mathchoice{\sbox 0{\displaystyle \perp \m@th }\sbox 2{\displaystyle =}\sbox 4{\displaystyle }}{\unknown. }{\copy 0}{}=-.2pt\hspace{0.0pt}{}\hspace{0.0pt}\copy 0 }{\sbox 0{\textstyle \perp \m@th }\sbox 2{\textstyle =}\sbox 4{\textstyle }}\unknown. {\copy 0}\hspace{-0.2pt}{}\hspace{-0.2pt}\copy 0 \)\(\scriptstyle \perp \m@th \)\(\scriptstyle \perp \m@th \)\(\scriptscriptstyle \perp \m@th \)\(\scriptscriptstyle \perp \m@th \)\( indicates independence.Under Conditions 1 and 2, \) 0TYi\( is redundant for a fixed effect regression \cite {cook2010envelope}.The \)\(-envelope is uniquely defined to be the smallest subspace satisfying these conditions. Once the basis \)\( is obtained, the envelope estimator is obtained by projecting the ordinary least square estimator onto the estimated envelope space. \cite {cook2010envelope} showed that the envelope estimator can achieve efficiency gain over the OLS estimator. Following the definition in their paper, we define the variance of \) T0 YiXi\( as the material part variance, and \) T0 Yi\( as the immaterial part variance. The efficiency gain will be substantial if the variation of the immaterial part is relatively large as compared with that of the material part.\) The envelope methods have been developed in different settings, including response envelope [1]}, partial envelope [2]}, inner envelope [3]}, scaled envelope [4]}, predictor envelope [5]}, reduced rank envelope [6]}, simultaneous envelope [7]}, model-free envelope [8]}, and tensor envelope [9]}.
